2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99824Let G be any finitely generated infinite group. Denote by K(G) the FC-centre of G, i.e., the subgroup of all elements of G whose centralizers are of finite index in G. Let QI(G) denote the group of quasi-isometries of G with respect to word metric. We observe that the natural homomorphism from the group of automorphisms of G to QI(G) is a monomorphism only if K(G) equals the centre Z(G) of G. The converse holds if K(G)=Z(G) is torsion free. We apply this criterion to many interesting classes of groups.This is the corrected version. Published in J. Group Theory, 8 (2005), 515--522Group Theory20F65, 20F28, 20F67text